pith. machine review for the scientific record. sign in

arxiv: 2605.13147 · v1 · submitted 2026-05-13 · 🧮 math.DS

Recognition: unknown

On fixed point results in metric spaces for large triangle-perimeter contractions

Ovidiu Popescu

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:42 UTC · model grok-4.3

classification 🧮 math.DS MSC 47H10
keywords fixed point theoremmetric spacecontraction mappingtriangle perimeterlarge contractionBurton mappingPetrov mapping
0
0 comments X

The pith

Fixed point theorem for large triangle-perimeter contractions holds only under an auxiliary condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first shows that an existing claim about fixed points for large triangle-perimeter contractions is false by exhibiting a counterexample. It then proves that the result becomes valid once an auxiliary condition is added to the contraction mapping. The new setting properly contains the earlier theories of large contractions and of triangle-perimeter contractions, as demonstrated by a concrete example that satisfies the new hypotheses but none of the older ones.

Core claim

In a complete metric space, every large triangle-perimeter contraction that also satisfies the stated auxiliary condition possesses a unique fixed point.

What carries the argument

Large triangle-perimeter contraction equipped with an auxiliary condition that forces the successive iterates to form a Cauchy sequence.

If this is right

  • The corrected statement recovers the classical large-contraction theorem of Burton as a special case.
  • The same statement recovers the triangle-perimeter result of Petrov when the auxiliary condition holds automatically.
  • The new class of mappings is strictly larger than either predecessor, allowing fixed-point guarantees for maps excluded by both earlier theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar auxiliary restrictions may be needed when extending other perimeter-type contractions to non-complete spaces.
  • Iterative schemes based on these mappings could be used to construct approximate fixed points numerically when the auxiliary inequality can be checked directly.

Load-bearing premise

The auxiliary condition on the contraction must be satisfied; without it the fixed-point conclusion can fail.

What would settle it

A complete metric space together with a mapping that meets the large triangle-perimeter definition and the auxiliary condition yet possesses no fixed point.

read the original abstract

In this paper we introduce a corrected extension of Burton's theory of large contractions in the context of triangle-perimeter contractions introduced by Petrov. Combining these two lines of research, we prove a fixed point result for large triangle-perimeter contractions with an auxiliary assumption, which is of utmost importance. Firstly, we give a counterexample to the main result related to large triangle-perimeter contractions that currently exists in literature. Then, we prove that given an additional condition, a fixed point result for large triangle-perimeter contractions holds. Lastly, we illustrate with an example that this new framework is strictly broader than Burton's and Petrov's theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that an auxiliary condition on large triangle-perimeter contractions in metric spaces suffices for a fixed-point theorem to hold. It supports the claim by exhibiting a counterexample to the prior uncorrected result in the literature, stating and proving the corrected theorem, and providing an example showing that the new setting strictly contains both Burton's large contractions and Petrov's triangle-perimeter contractions.

Significance. If the corrected result holds, the work is significant for identifying and repairing an error in the literature on generalized contractions. The explicit counterexample and the concrete example establishing strict inclusion are clear strengths that advance the understanding of the relationship between these contraction classes. The proof follows standard contraction-mapping arguments once the auxiliary condition is imposed.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'of utmost importance' is subjective and should be replaced by a precise statement such as 'necessary for the fixed-point conclusion to hold under the given hypotheses'.
  2. [§3] §3 (Theorem statement): the auxiliary condition is introduced without a numbered label or displayed equation; assigning it a label such as (AC) would improve cross-referencing in the proof.
  3. [§4] §4 (Example): the metric d on the space X is described only verbally; an explicit formula for d(x,y) should be supplied so that the verification of the large triangle-perimeter property can be checked directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of the counterexample to the prior claim and the example establishing strict inclusion as clear strengths, and the recommendation for minor revision. We will incorporate any minor changes needed to finalize the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation begins from the definitions of large contractions and triangle-perimeter contractions, exhibits a counterexample to a prior claim, adds an explicit auxiliary condition, and proves the fixed-point result directly from the metric axioms plus that condition. No step reduces a prediction to a fitted input, renames a known result, or relies on a self-citation chain whose justification is internal to the present work. The central theorem is therefore self-contained against the stated hypotheses and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard axioms of metric spaces and the definitions of large contractions and triangle-perimeter contractions from Burton and Petrov. No free parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption The underlying space is a complete metric space.
    Required for the contraction mapping principle to guarantee convergence to a fixed point.
  • domain assumption Triangle-perimeter contraction as defined by Petrov.
    The central object of study is taken directly from the cited prior work.

pith-pipeline@v0.9.0 · 5391 in / 1246 out tokens · 58362 ms · 2026-05-14T18:42:14.424731+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

8 extracted references

  1. [1]

    Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922), 133-181

    S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales, Fundamenta Mathematicae 3 (1922), 133-181

    1922

  2. [2]

    Burton, Integral equations, implicit functions, and fixed points

    T.A. Burton, Integral equations, implicit functions, and fixed points. Proc. Am. Math. Soc. 1996, 124, 2383–2390

    1996

  3. [3]

    S. K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727–730

    1972

  4. [4]

    Dehici, M.B

    A. Dehici, M.B. Mesmouli, E. Karapinar, On the fixed points of large- Kannan contraction mappings and applications. Appl. Math.-E-Notes 2019, 19, 535–551. 9

    2019

  5. [5]

    Kannan, Some results on fixed point - II, Amer

    R. Kannan, Some results on fixed point - II, Amer. Math. Monthly, 76 (1969), 405-408

    1969

  6. [6]

    Mesmouli, E

    M.B. Mesmouli, E. Akın, L.F. Iambor, O. Tun¸ c, T.S. Hassan, On the fixed point theorem for large contraction mappings with applications to delay frac- tional differential equations. Fractal Fract. 2024, 8, 703

    2024

  7. [7]

    Mesmouli, L.F

    M.B. Mesmouli, L.F. Iambor, T.S. Hassan, Fixed-Point Results in Metric Spaces for Large Triangle–Perimeter Contractions. Mathematics 2026, 14, 457

    2026

  8. [8]

    Petrov, Fixed point theorem for mappings contracting perimeters of trian- gles

    E. Petrov, Fixed point theorem for mappings contracting perimeters of trian- gles. J. Fixed Point Theory Appl. 2023, 25, 74. 10

    2023